## Kicked-Harper model vs On-Resonance Double Kicked Rotor Model: From Spectral Difference to Topological Equivalence    [PDF]

Hailong Wang, Derek Y. H. Ho, Wayne Lawton, Jiao Wang, Jiangbin Gong
Recent studies have established that, in addition to the well-known kicked Harper model (KHM), an on-resonance double kicked rotor model (ORDKR) also has Hofstadter's butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter's spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that these quasi-energy spectra of the two dynamical models can (i) exactly overlap with each other if an effective Planck constant takes irrational multiples of $2\pi$ and (ii) will be different if the same parameter takes rational multiples of $2\pi$. This work makes some detailed comparison between these two models, with an effective Planck constant given by $2\pi M/N$, where $M$ and $N$ are coprime integers. It is found that for odd $M$ and $N$, the ORDKR spectrum has one flat band and $N-1$ non-flat bands whose widths decay in power law as $\sim K^{N+2}$, where $K$ is a kicking strength parameter. The existence of a flat band is strictly proved and the power law scaling, numerically checked for a number of cases, is also analytically proved for a three-band case. By contrast, the KHM does not have any flat band and their band width scales linearly with $K$. This is shown to result in dramatic differences in their dynamical behavior, such as transient (but extremely long) dynamical localization in ORDKR. Finally, we show that despite these differences, from a topological point of view, KHM and ORDRK are actually topologically equivalent. A theoretical derivation of this topological equivalence is provided.
View original: http://arxiv.org/abs/1306.6128